The concept of evenness or oddness is only defined for functions whose domain and range both have an additive inverse. This includes additive groups, all rings, all fields, and all vector spaces. Thus, for example, a real-valued function of a real variable could be even or odd, as could a complex-valued function of a vector variable, and so on.

The examples are real-valued functions of a real variable, to illustrate the symmetry of their graphs.

A function's being odd or even does not imply differentiability, or even continuity. For example, the Dirichlet function is even, but is nowhere continuous. Properties involving Fourier series, Taylor series, derivatives and so on may only be used when they can be assumed to exist.

Any linear combination of even functions is even, and the even functions form a vector space over the reals. Similarly, any linear combination of odd functions is odd, and the odd functions also form a vector space over the reals. In fact, the vector space of all real-valued functions is the direct sum of the subspaces of even and odd functions. In other words, every function f(x) can be written uniquely as the sum of an even function and an odd function:

where

is even and

is odd. For example, if f is exp, then fe is cosh and fo is sinh.

The even functions form a commutative algebra over the reals. However, the odd functions do not form an algebra over the reals, as they are not closed under multiplication.

The integral of an odd function from −A to +A is zero (where A is finite, and the function has no vertical asymptotes between −A and A).

The integral of an even function from −A to +A is twice the integral from 0 to +A (where A is finite, and the function has no vertical asymptotes between −A and A. This also holds true when A is infinite, but only if the integral converges).

In signal processing, harmonic distortion occurs when a sine wave signal is sent through a memoryless nonlinear system, that is, a system whose output at time only depends on the input at time and does not depend on the input at any previous times. Such a system is described by a response function . The type of harmonics produced depend on the response function :[3]

When the response function is even, the resulting signal will consist of only even harmonics of the input sine wave;

When it is asymmetric, the resulting signal may contain either even or odd harmonics;

Simple examples are a half-wave rectifier, and clipping in an asymmetrical class-A amplifier.

Note that this does not hold true for more complex waveforms. A sawtooth wave contains both even and odd harmonics, for instance. After even-symmetric full-wave rectification, it becomes a triangle wave, which, other than the DC offset, contains only odd harmonics.